Abstract

We establish some oscillation criteria for the following certain even order neutral delay differential equations with mixed nonlinearities: where is even integer, and Our results generalize and improve some known results for oscillation of certain even order neutral delay differential equations with mixed nonlinearities.

1. Introduction

In this paper, we are concerned with oscillation behavior of the certain even order neutral delay differential equations with mixed nonlinearities:
where , is even integer, and are constants. , , satisfy that , , and there exists a function , such that , . We assume that there exists a function , such that , , , , and .

We will consider the two cases

Recently, there have been a large number of papers devoted to the oscillation of the delay differential equations; see [1–8]. Furthermore, there have been a large number of works on the oscillation of the neutral differential equations, and we refer the readers to the articles [9–25].

Agarwal and Grace [3] studied the oscillation for functional differential equations of higher order,
and established some sufficient conditions for oscillation of (4).

Han et al. [9] studied the oscillation of second-order neutral differential equations:
where and , . Some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations:
where , .

Meng and Xu [19], by using the Riccati transformation technique and inequalities, considered the oscillation for even order quasilinear neutral differential equations:
where , .

In 2012, Sun et al. [22] considered the oscillation criteria for even order nonlinear neutral differential equations:
where , is even integer, and . The results are obtained when or . These criteria obtained in this paper extended and improved some known results in the literatures.

In 2013, Agarwal et al. [24] considered the oscillation criteria for even order neutral differential equations:
Some new criteria are established that improve a number of related results reported in the literature and can be used in cases where known theorems fail to apply.

In 2014, Zhang et al. [25] study oscillation and asymptotic behavior of solutions to two classes of higher-order delay damped differential equations with -Laplacian like operators:
where . Some new criteria are presented that improve the related contributions to the subject.

Clearly, the equations (4)–(12) are special cases of (1). The purpose of this paper is to extend and improve the abovementioned oscillation theorems for certain even order neutral delay differential equations with mixed nonlinearities (1).

The paper is organized as follows. In the next section, we present some lemmas which will be used in the following results. In Sections 3 and 4, by developing Riccati transformations technique and inequalities, some sufficient conditions for oscillation of all solutions of (1) are established. In Section 5, we give an example to illustrate Theorem 11.

2. Lemmas

In this section, in order to prove our main results, we need the following lemmas.

Lemma 1 (see [5]). Let . If is eventually of one sign for all large , then there exist , for some , and an integer , , with even for or odd for such that implies that for , , and implies that for , .

Lemma 2 (see [1], Lemma ). If the function is as in Lemma 1 and for , then, for every , , there exists a constant such that
for all large .

Lemma 3 (see [1], Lemma ). If the function is as in Lemma 1 and for , , then, for every , , such that
for all large .

Lemma 4 (see [2, 5]). Consider the half-linear differential equations
where , , . Then every solution of (15) is nonoscillatory if and only if there exist a real number and a function , such that

Lemma 5. If and are nonnegative constants, then
and the equality holds if and only if .

In the next section, by developing Riccati transformations technique and inequalities, some sufficient conditions for oscillation of all solutions of (1) are established.

3. Oscillation Criteria for an Oscillating Function

In this section, we assume the following.

(H) is an oscillating function, and .

Lemma 6. Assume that (2) holds. Furthermore, assume that is an eventually positive solution of (1), which is bounded and does not converge to zero. Then there exists , such that

Proof. Since is an eventually positive solution of (1), there exists a constant , such that , , and , , for all . Then, by (1), we have , .Furthermore, since is a bounded solution and , by (H) we know that ; then there exists , such that , . So is eventually positive and bounded.The rest of the proof is similar to that of Meng and Xu [19, Lemma 2.3], so it is omitted.

Theorem 7. Assume that (H) and (2) hold. Furthermore, assume that there exists a constant , , and, for every constant , assume that there exists a positive function , for sufficiently large , such that
where
Then every bounded solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a bounded nonoscillatory solution . We may assume without loss of generality that there exists a number , such that , , and , for all . Furthermore, we assume that . Using the definition of and Lemma 6, we have , , , and , . Hence there exists , such that
From (1) and the above inequality, we obtain
Because of , by Lemma 2, , and , there exists , and, for every , there exists a constant , we have
for . We define the function by
Then , . Next differentiating (24), we get
So by (22) and (23), we obtain
Let
where and are defined as in Theorem 7. Using the inequality
we have
so we get
Let
where . Applying the inequality in Lemma 5, we obtain
Thus, by (30) and (32), we get
Integrating (33) from to , we have
Let in (34), which leads to a contradiction with (19). The proof is complete.

Theorem 8. Assume that (H) and (2) hold, and there exists a constant , , and, for every constant , such that
is oscillatory, where is defined as in Theorem 7. Then every bounded solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a bounded nonoscillatory solution . We may assume without loss of generality that there exists , such that , , and , for all . Furthermore, we assume that . We define by
Proceeding as in the proof of Theorem 7, for every , there exists , and we have
That is,
Based on Lemma 4, we obtain that (35) is nonoscillatory, which leads to a contradiction. The proof is complete.

Theorem 9. Assume that (H) and (3) hold. Furthermore, assume that there exists a constant , , and, for every constant , assume that there exists a positive function , such that (19) holds. If, for sufficiently large ,
where and are defined as in Theorem 7, and then every bounded solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a bounded nonoscillatory solution . We may assume without loss of generality that there exists , such that , , and , for all . Then it follows from (1) that
Therefore, is a nonincreasing function on . Consequently, it is easy to conclude that there exist two possible cases of the sign of . Furthermore, we assume that .Case I. If , for , then we go back to the proof of Theorem 7, and we get a contradiction to (19), so we omit the details.Case II. , for . Applying Lemma 1, we get . Define the function by
Then for . Noting that is nonincreasing, we obtain
Dividing (42) by and integrating it from to , we have
Letting in the above inequality, we get
which implies that
where is defined as in Theorem 9. Hence, by (41), we obtain
Differentiating (41), we have
From (22), we get
On the other hand, by and Lemma 3, we obtain
that is, because of ,
for every and . Then from (41), (48), and (50), we have
where is defined as in Theorem 7. Multiplying (51) by and integrating it from to , we get
Let
where . Applying the inequality in Lemma 5, we obtain
Therefore, it follows from (52) that
From (39) and the above inequality, we get a contradiction to (46). The proof is complete.

Theorem 10. Assume that (H) and (3) hold. Furthermore, assume that there exists a constant , , and for every constant such that (35) is oscillatory. If, for sufficiently large , one has (39), where and are defined as in Theorem 7 and is defined as in Theorem 9, then every bounded solution of (1) is oscillatory or converges to zero.

4. Oscillation Criteria for

In this section, we assume that .

Theorem 11. Assume that (2) holds, there exists a constant , , and, for every constant , there exists a positive function , such that, for sufficiently large ,
where
and and are defined as in Theorem 7. Then every solution of (1) is oscillatory.

Proof. Suppose that (1) has a nonoscillatory solution . Without loss of generality, we may assume that there exists , such that , , and , for all . Similar to the proof of Lemma 2.3 in [19], there exists , such that
From the definition of , we have
Since , there exists , such that , , so
From (1), (59), and (60), we get
For every , we define the function
Then , . Next differentiating (62), we obtain
From (23), (61), and (62), we have
where is defined as in Theorem 11. Setting
by inequality (28), we get
hence,
Let
where . Applying the inequality in Lemma 5, we obtain
Thus, by (67) and (69), we get
Integrating (70) from to , we have
Letting in (71), we get a contradiction with (56). This completes the proof of Theorem 11.

Remark 12. From Theorem 11, we can obtain different conditions for oscillation of all solutions of (1) with different choices of .

Theorem 13. Assume that (3) holds, assume that there exists a constant , , and, for every constant , there exists a positive function , such that, for sufficiently large , (56) holds. If there exists a positive function , , such that
where is defined as in Theorem 9 and is defined as in Theorem 11, then every solution of (1) is oscillatory or converges to zero.

Proof. Suppose that (1) has a nonoscillatory solution . We may assume without loss of generality that there exists , such that , , , and , for all . Furthermore, we assume that . Similar to the proof of Theorem 9, we find that is a nonincreasing function on and there exist two possible cases of the sign of .Case I. If , for , then we go back to the proof of Theorem 11, and we get a contradiction to (56), so we omit the details.Case II. , for . Applying Lemma 1, we get , or , .If , . Since is nonincreasing, we obtain
for ; that is,
Integrating (75) from to , we get
where . From (49) and the above inequality, we obtain
Using (59) and (77) in (1) and noting that , we have
Setting
by inequality (28), we get
Therefore, combining (77), (78), and (80), we obtain
Define the function by
Then . Differentiating and from (81), we find that
Integrating (83) from to , we get
Therefore,
that is,
Integrating the above inequality from to , we obtain
Letting and using (73) in (97), we have , which is a contradiction with the fact that .If , . Because of , . By Lemma 3, we obtain (49). Proceeding as in the proof of the above, (77) holds. Using (59) and (77) in (1) and noting that , we have
Setting
by inequality (28), we get
Therefore, combining (77), (88), and (90), we obtain
Define the function by
Then
Integrating from to , we get